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  <section id="module-sympy.combinatorics.named_groups">
<span id="named-groups"></span><span id="combinatorics-named-groups"></span><h1>Named Groups<a class="headerlink" href="#module-sympy.combinatorics.named_groups" title="Permalink to this headline">¶</a></h1>
<dl class="py function">
<dt class="sig sig-object py" id="sympy.combinatorics.named_groups.SymmetricGroup">
<span class="sig-prename descclassname"><span class="pre">sympy.combinatorics.named_groups.</span></span><span class="sig-name descname"><span class="pre">SymmetricGroup</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/combinatorics/named_groups.py#L241-L305"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.SymmetricGroup" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates the symmetric group on <code class="docutils literal notranslate"><span class="pre">n</span></code> elements as a permutation group.</p>
<p class="rubric">Explanation</p>
<p>The generators taken are the <code class="docutils literal notranslate"><span class="pre">n</span></code>-cycle
<code class="docutils literal notranslate"><span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></code> and the transposition <code class="docutils literal notranslate"><span class="pre">(0</span> <span class="pre">1)</span></code> (in cycle notation).
(See [1]). After the group is generated, some of its basic properties
are set.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">SymmetricGroup</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">SymmetricGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
<span class="go">24</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate_schreier_sims</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
<span class="go">[[0, 1, 2, 3], [1, 2, 3, 0], [2, 3, 0, 1], [3, 1, 2, 0], [0, 2, 3, 1],</span>
<span class="go">[1, 3, 0, 2], [2, 0, 1, 3], [3, 2, 0, 1], [0, 3, 1, 2], [1, 0, 2, 3],</span>
<span class="go">[2, 1, 3, 0], [3, 0, 1, 2], [0, 1, 3, 2], [1, 2, 0, 3], [2, 3, 1, 0],</span>
<span class="go">[3, 1, 0, 2], [0, 2, 1, 3], [1, 3, 2, 0], [2, 0, 3, 1], [3, 2, 1, 0],</span>
<span class="go">[0, 3, 2, 1], [1, 0, 3, 2], [2, 1, 0, 3], [3, 0, 2, 1]]</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#sympy.combinatorics.named_groups.CyclicGroup" title="sympy.combinatorics.named_groups.CyclicGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">CyclicGroup</span></code></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.DihedralGroup" title="sympy.combinatorics.named_groups.DihedralGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">DihedralGroup</span></code></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.AlternatingGroup" title="sympy.combinatorics.named_groups.AlternatingGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">AlternatingGroup</span></code></a></p>
</div>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r42"><span class="brackets"><a class="fn-backref" href="#id1">R42</a></span></dt>
<dd><p><a class="reference external" href="https://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations">https://en.wikipedia.org/wiki/Symmetric_group#Generators_and_relations</a></p>
</dd>
</dl>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="sympy.combinatorics.named_groups.CyclicGroup">
<span class="sig-prename descclassname"><span class="pre">sympy.combinatorics.named_groups.</span></span><span class="sig-name descname"><span class="pre">CyclicGroup</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/combinatorics/named_groups.py#L130-L171"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.CyclicGroup" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates the cyclic group of order <code class="docutils literal notranslate"><span class="pre">n</span></code> as a permutation group.</p>
<p class="rubric">Explanation</p>
<p>The generator taken is the <code class="docutils literal notranslate"><span class="pre">n</span></code>-cycle <code class="docutils literal notranslate"><span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></code>
(in cycle notation). After the group is generated, some of its basic
properties are set.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">CyclicGroup</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">CyclicGroup</span><span class="p">(</span><span class="mi">6</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">order</span><span class="p">()</span>
<span class="go">6</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate_schreier_sims</span><span class="p">(</span><span class="n">af</span><span class="o">=</span><span class="kc">True</span><span class="p">))</span>
<span class="go">[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1],</span>
<span class="go">[3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]]</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#sympy.combinatorics.named_groups.SymmetricGroup" title="sympy.combinatorics.named_groups.SymmetricGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SymmetricGroup</span></code></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.DihedralGroup" title="sympy.combinatorics.named_groups.DihedralGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">DihedralGroup</span></code></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.AlternatingGroup" title="sympy.combinatorics.named_groups.AlternatingGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">AlternatingGroup</span></code></a></p>
</div>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="sympy.combinatorics.named_groups.DihedralGroup">
<span class="sig-prename descclassname"><span class="pre">sympy.combinatorics.named_groups.</span></span><span class="sig-name descname"><span class="pre">DihedralGroup</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/combinatorics/named_groups.py#L174-L238"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.DihedralGroup" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates the dihedral group <span class="math notranslate nohighlight">\(D_n\)</span> as a permutation group.</p>
<p class="rubric">Explanation</p>
<p>The dihedral group <span class="math notranslate nohighlight">\(D_n\)</span> is the group of symmetries of the regular
<code class="docutils literal notranslate"><span class="pre">n</span></code>-gon. The generators taken are the <code class="docutils literal notranslate"><span class="pre">n</span></code>-cycle <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">=</span> <span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></code>
(a rotation of the <code class="docutils literal notranslate"><span class="pre">n</span></code>-gon) and <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">=</span> <span class="pre">(0</span> <span class="pre">n-1)(1</span> <span class="pre">n-2)...</span></code>
(a reflection of the <code class="docutils literal notranslate"><span class="pre">n</span></code>-gon) in cycle rotation. It is easy to see that
these satisfy <code class="docutils literal notranslate"><span class="pre">a**n</span> <span class="pre">=</span> <span class="pre">b**2</span> <span class="pre">=</span> <span class="pre">1</span></code> and <code class="docutils literal notranslate"><span class="pre">bab</span> <span class="pre">=</span> <span class="pre">~a</span></code> so they indeed generate
<span class="math notranslate nohighlight">\(D_n\)</span> (See [1]). After the group is generated, some of its basic properties
are set.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">DihedralGroup</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">DihedralGroup</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate_dimino</span><span class="p">())</span>
<span class="gp">&gt;&gt;&gt; </span><span class="p">[</span><span class="n">perm</span><span class="o">.</span><span class="n">cyclic_form</span> <span class="k">for</span> <span class="n">perm</span> <span class="ow">in</span> <span class="n">a</span><span class="p">]</span>
<span class="go">[[], [[0, 1, 2, 3, 4]], [[0, 2, 4, 1, 3]],</span>
<span class="go">[[0, 3, 1, 4, 2]], [[0, 4, 3, 2, 1]], [[0, 4], [1, 3]],</span>
<span class="go">[[1, 4], [2, 3]], [[0, 1], [2, 4]], [[0, 2], [3, 4]],</span>
<span class="go">[[0, 3], [1, 2]]]</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#sympy.combinatorics.named_groups.SymmetricGroup" title="sympy.combinatorics.named_groups.SymmetricGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SymmetricGroup</span></code></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.CyclicGroup" title="sympy.combinatorics.named_groups.CyclicGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">CyclicGroup</span></code></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.AlternatingGroup" title="sympy.combinatorics.named_groups.AlternatingGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">AlternatingGroup</span></code></a></p>
</div>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r43"><span class="brackets"><a class="fn-backref" href="#id2">R43</a></span></dt>
<dd><p><a class="reference external" href="https://en.wikipedia.org/wiki/Dihedral_group">https://en.wikipedia.org/wiki/Dihedral_group</a></p>
</dd>
</dl>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="sympy.combinatorics.named_groups.AlternatingGroup">
<span class="sig-prename descclassname"><span class="pre">sympy.combinatorics.named_groups.</span></span><span class="sig-name descname"><span class="pre">AlternatingGroup</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/combinatorics/named_groups.py#L56-L127"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.AlternatingGroup" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates the alternating group on <code class="docutils literal notranslate"><span class="pre">n</span></code> elements as a permutation group.</p>
<p class="rubric">Explanation</p>
<p>For <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">&gt;</span> <span class="pre">2</span></code>, the generators taken are <code class="docutils literal notranslate"><span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2),</span> <span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></code> for
<code class="docutils literal notranslate"><span class="pre">n</span></code> odd
and <code class="docutils literal notranslate"><span class="pre">(0</span> <span class="pre">1</span> <span class="pre">2),</span> <span class="pre">(1</span> <span class="pre">2</span> <span class="pre">...</span> <span class="pre">n-1)</span></code> for <code class="docutils literal notranslate"><span class="pre">n</span></code> even (See [1], p.31, ex.6.9.).
After the group is generated, some of its basic properties are set.
The cases <code class="docutils literal notranslate"><span class="pre">n</span> <span class="pre">=</span> <span class="pre">1,</span> <span class="pre">2</span></code> are handled separately.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">AlternatingGroup</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span> <span class="o">=</span> <span class="n">AlternatingGroup</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">G</span><span class="o">.</span><span class="n">is_group</span>
<span class="go">True</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">a</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">G</span><span class="o">.</span><span class="n">generate_dimino</span><span class="p">())</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">len</span><span class="p">(</span><span class="n">a</span><span class="p">)</span>
<span class="go">12</span>
<span class="gp">&gt;&gt;&gt; </span><span class="nb">all</span><span class="p">(</span><span class="n">perm</span><span class="o">.</span><span class="n">is_even</span> <span class="k">for</span> <span class="n">perm</span> <span class="ow">in</span> <span class="n">a</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="#sympy.combinatorics.named_groups.SymmetricGroup" title="sympy.combinatorics.named_groups.SymmetricGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">SymmetricGroup</span></code></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.CyclicGroup" title="sympy.combinatorics.named_groups.CyclicGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">CyclicGroup</span></code></a>, <a class="reference internal" href="#sympy.combinatorics.named_groups.DihedralGroup" title="sympy.combinatorics.named_groups.DihedralGroup"><code class="xref py py-obj docutils literal notranslate"><span class="pre">DihedralGroup</span></code></a></p>
</div>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r44"><span class="brackets"><a class="fn-backref" href="#id3">R44</a></span></dt>
<dd><p>Armstrong, M. “Groups and Symmetry”</p>
</dd>
</dl>
</dd></dl>

<dl class="py function">
<dt class="sig sig-object py" id="sympy.combinatorics.named_groups.AbelianGroup">
<span class="sig-prename descclassname"><span class="pre">sympy.combinatorics.named_groups.</span></span><span class="sig-name descname"><span class="pre">AbelianGroup</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">cyclic_orders</span></span></em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/sympy/sympy/blob/00d6469eafdd4aac346a0b598184c15f2560dbe5/sympy/combinatorics/named_groups.py#L8-L53"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#sympy.combinatorics.named_groups.AbelianGroup" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the direct product of cyclic groups with the given orders.</p>
<p class="rubric">Explanation</p>
<p>According to the structure theorem for finite abelian groups ([1]),
every finite abelian group can be written as the direct product of
finitely many cyclic groups.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="kn">from</span> <span class="nn">sympy.combinatorics.named_groups</span> <span class="kn">import</span> <span class="n">AbelianGroup</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">AbelianGroup</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="go">PermutationGroup([</span>
<span class="go">        (6)(0 1 2),</span>
<span class="go">        (3 4 5 6)])</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">_</span><span class="o">.</span><span class="n">is_group</span>
<span class="go">True</span>
</pre></div>
</div>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<p><a class="reference internal" href="group_constructs.html#sympy.combinatorics.group_constructs.DirectProduct" title="sympy.combinatorics.group_constructs.DirectProduct"><code class="xref py py-obj docutils literal notranslate"><span class="pre">DirectProduct</span></code></a></p>
</div>
<p class="rubric">References</p>
<dl class="citation">
<dt class="label" id="r45"><span class="brackets"><a class="fn-backref" href="#id4">R45</a></span></dt>
<dd><p><a class="reference external" href="http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups">http://groupprops.subwiki.org/wiki/Structure_theorem_for_finitely_generated_abelian_groups</a></p>
</dd>
</dl>
</dd></dl>

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